Spatiospectral concentration on a sphere

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.Comment: 48 pages, 17 figures. Submitted to SIAM Review, August 24th, 200

Use of remotely-derived bathymetry for modelling biomass in marine environments

The paper presents results on the influence of geometric attributes of satellite-derived raster bathymetric data, namely the General Bathymetric Charts of the Oceans, on spatial statistical modelling of marine biomass. In the initial experiment, both the resolution and projection of the raster dataset are taken into account. It was found that, independently of the equal-area projection chosen for the analysis, the calculated areas are very similar, and the differences between them are insignificant. Likewise, any variation in the raster resolution did not change the computed area. Although the differences were shown to be insignificant, for the subsequent analysis we selected the cylindrical equal area projection, as it implies rectangular spatial extent, along with the automatically derived resolution. Then, in the second experiment, we focused on demersal fish biomass data acquired from trawl samples taken from the western parts of ICES Sub-area VII, near the sea floor. The aforementioned investigation into processing bathymetric data allowed us to build various statistical models that account for a relationship between biomass, sea floor topography and geographic location. We fitted a set of generalised additive models and generalised additive mixed models to combinations of trawl data of the roundnose grenadier (Coryphaenoides rupestris) and bathymetry. Using standard statistical techniques—such as analysis of variance, Akaike information criterion, root mean squared error, mean absolute error and cross-validation—we compared the performance of the models and found that depth and latitude may serve as statistically significant explanatory variables for biomass of roundnose grenadier in the study area. However, the results should be interpreted with caution as sampling locations may have an impact on the biomass–depth relationship

How large are present-day heat flux variations across the surface of Mars?

©2016. American Geophysical UnionThe first in situ Martian heat flux measurement to be carried out by the InSight Discovery‐class mission will provide an important baseline to constrain the present‐day heat budget of the planet and, in turn, the thermochemical evolution of its interior. In this study, we estimate the magnitude of surface heat flux heterogeneities in order to assess how the heat flux at the InSight landing site relates to the average heat flux of Mars. To this end, we model the thermal evolution of Mars in a 3‐D spherical geometry and investigate the resulting surface spatial variations of heat flux at the present day. Our models assume a fixed crust with a variable thickness as inferred from gravity and topography data and with radiogenic heat sources as obtained from gamma ray measurements of the surface. We test several mantle parameters and show that the present‐day surface heat flux pattern is dominated by the imposed crustal structure. The largest surface heat flux peak‐to peak variations lie between 17.2 and 49.9 mW m−2, with the highest values being associated with the occurrence of prominent mantle plumes. However, strong spatial variations introduced by such plumes remain narrowly confined to a few geographical regions and are unlikely to bias the InSight heat flux measurement. We estimated that the average surface heat flux varies between 23.2 and 27.3 mW m−2, while at the InSight location it lies between 18.8 and 24.2 mW m−2. In most models, elastic lithosphere thickness values exceed 250 km at the north pole, while the south pole values lie well above 110 km

Advancing a Democratic Pedagogy and Supervision Framework: An Illustrative Case of Teacher Questioning in Secondary Mathematics Instruction

This article pushes back against the evalu-centric view of improvement (Hazi, 2018; 2020) in the supervision literature by advocating for a democratic pedagogy and supervision framework developed to support instructional supervision and evaluation dialogue between teachers and leaders. This democratized approach honors and centers the teacher’s expertise and learning as well as the leader’s in the observation, debrief, and reflection process. Through this decentering of expertise in the instructional supervision cycle, our goal is to build leaders’ and teachers’ mutual capacity to develop, implement, and sustain democratic instructional supervision cultures in classrooms and schools. Additionally, we illustrate our framework through a subject/discipline-specific case of instructional supervision in secondary mathematics instruction. Through this illustrative case, we demonstrate how the framework provides school leaders and teachers with specific, shared pedagogical language to engage in standards-based mathematical dialogue during the instructional supervision process. Finally, we discuss the implications of our questioning framework for democratic school leadership, supervisors’ leadership content knowledge, teachers’ discipline-specific work of teaching, and instructional supervision practices, which are often stifled by accountability-driven teacher evaluation education policies that suppress schools’ leadership capacity to apply democratic instructional supervision standards and principles

Mercury's low‐degree geoid and topography controlled by insolation‐driven elastic deformation

©2015. American Geophysical UnionMercury experiences an uneven insolation that leads to significant latitudinal and longitudinal variations of its surface temperature. These variations, which are predominantly of spherical harmonic degrees 2 and 4, propagate to depth, imposing a long‐wavelength thermal perturbation throughout the mantle. We computed the accompanying density distribution and used it to calculate the mechanical and gravitational response of a spherical elastic shell overlying a quasi‐hydrostatic mantle. We then compared the resulting geoid and surface deformation at degrees 2 and 4 with Mercury's geoid and topography derived from the MErcury, Surface, Space ENvironment, GEochemistry, and Ranging spacecraft. More than 95% of the data can be accounted for if the thickness of the elastic lithosphere were between 110 and 180 km when the thermal anomaly was imposed. The obtained elastic thickness implies that Mercury became locked into its present 3:2 spin orbit resonance later than about 1 Gyr after planetary formation

Nonuniform cratering of the Moon and a revised crater chronology of the inner solar system

International audience► We model the cratering of the Moon and terrestrial planets. ► We account for cratering asymmetries and megaregolith. ► We revise the crater chronology method. ► We give new age estimates of key planetary surface

Ejecta deposit thickness, heat flow, and a critical ambiguity on the Moon

The Apollo lunar heat flow measurements gave values of 21 and 16 mW m?2 which, after extrapolation based on thorium abundances, yields a global estimate of 18 mW m?2. A refinement of the assumptions of the subsurface structure and the resulting focusing of heat flux later led to a revision of the global value to 12 mW m?2. We think that to date none of the models linking the Apollo heat flow measurements has sufficiently highlighted a critical source of ambiguity. Little attention has been paid to the full magnitude of the uncertainty in these measurements caused by near–surface Thorium abundances and the local thickness of the ejecta blanket generated by the Imbrium impact. In a simple study we show that lunar heat flow is contingent upon the thickness of the ejecta blanket of the hypothetical impact. A model with an exponential decrease of Th concentration with depth can explain the difference in surface heat flow between the Apollo 15 and the Apollo 17 measurements. A constant Thorium concentration within the ejecta layer amplifies this effect. The variation in local surface Th abundance, if taken as representative of the subsurface Th distribution within the ejecta blanket, amplifies the uncertainty. We conclude that further measurements are essential for making well-founded statements about the subsurface abundance of radioactive elements, mantle heat flux and the thermal state of the Moon